Expected Utility Theory with Probability Grids and. Preference Formation

Transcription

1 WINPEC Working Paper Series No.E1902 April 2019 Expected Utility Theory with Probability Grids and Preference Formation Mamoru Kaneko Waseda INstitute of Political EConomy Waseda University Tokyo, Japan

2 Expected Utility Theory with Probability Grids and Preference Formation Mamoru Kaneko y March 30, 2019 Abstract We reformulate expected utility theory, from the viewpoint of bounded rationality, by introducing probability grids and a cognitive bound; we restrict permissible probabilities only to decimal (`-ary in general) fractions of nite depths up to a given cognitive bound. We distinguish between measurements of utilities from pure alternatives and their extensions to lotteries involving more risks. Our theory is constructive, from the viewpoint of the decision maker. When a cognitive bound is small, the preference relation involves many incomparabilities, but these diminish as the cognitive bound is is relaxed. Similarly, the EU hypothesis would hold more for a weaker cognitive bound. The main part of the paper is a study of preferences including incomparabilities in cases with nite cogntive bounds; we give representation theorems in terms of a 2-dimensional vector-valued utility functions. We exemplify the theory with one experimental result reported by Kahneman-Tversky. JEL Classi cation Numbers: C72, C79, C91 Key Words: Expected Utility, Measurement of Utility, Bounded Rationality, Probability Grids, Cognitive Bound, Incomparabilities 1 Introduction We reconsider EU theory from the viewpoint of preference formation and of bounded rationality. We restrict permissible probabilities to decimal (`-ary, in general) fractions up to a given cognitive bound ; if is a natural number k, the set of permissible probabilities is given as = k = f 0 1 ; ; :::; k g: The decision maker makes preference comparisons step by step k k k using probabilities with small k to those with larger k 0 to obtain accurate comparisons. The derived preference relation is incomplete in general, but the EU hypothesis holds for some lotteries and would hold more when there is no cognitive bound, i.e., = 1: Our main concern is the case < 1. Since the theory involves various entangled aspects, we rst disentangle them. The concepts of probability grids and cognitive bounds are introduced based on the idea of bounded rationality. This idea can be interpreted in many ways such as bounded logical The author thanks J. J. Kline, P. Wakker, M. Lewandowski, S. Shiba, M. Cohen, O. Shulte, and Y. Rebille for helpful comments on earlier versions of this paper. In particular, comments given by the two referees improved the paper signi cantly and are greatly appreciated. The author is supported by Grant-in-Aids for Scienti c Research No and No.17H02258, Ministry of Education, Science and Culture. y Waseda University, Tokyo, Japan (mkanekoepi@waseda.jp) 1

3 inference, bounded perception ability, though Simon s [29] original concept meant a relaxation of utility maximization. The mathematical components involved in EU theory are classi ed to two types; object-components used by the decision maker and meta-components used by the outside analyst and possibly by the decision maker himself. The former is primary targets in EU theory, and the latter such as highly complex rational as well as irrational probabilities is added for analytic convenience. A free use of the latter leads to a critique that the theory presumes super rationality (Simon [30]). As a signi cance level for statistical hypothesis testing is typically 5% or 1%; probability values (t = 0; :::; 2 ) are already quite accurate for ordinary people. However, the classical EU t 2 theory starts with the full real number theory and makes no separation between the viewpoints of the decision maker and the outside analyst for available probabilities. This is still a problem of degree, but it would be meaningful if they are separated in some manner. The concepts of probability grids and a cognitive bound make this separation. The set of probability grids up to depth k is given as k = f 0 1 ; ; :::; k g. The decision k k k maker thinks about his preferences with k from a small k to a larger k up to bound ; for example, when = 2; 0 ; 1 ; and 2 are only allowed. This is a constructive approach from the viewpoint of the decision maker in the sense that he nds/forms his own preferences. 1;2 We turn our attention to the development of our constructive EU theory. Constructiveness needs a start; we take a hint from von Neumann-Morgenstern [31]. They divided the motivating argument into the following two, though this separation was not re ected in their development: Step B: measurements of utilities from pure alternatives in terms of probabilities; Step E: extensions of these measurements to lotteries involving more risks. These steps di er in their natures: Step B is to measure a satisfaction, desire, etc. from a pure alternative, while Step E is to extend the measured satisfactions given by Step B to lotteries including more risks. An important di erence is that Step B is to nd the subjective preferences hidden in the mind of the decision maker, while Step E is to extend logically the preferences found in Step B to lotteries with more risks. We develop our theory based on the above two steps and also take two approaches in terms of preferences and utilities; each approach consists of Steps B and E. In this introduction, we focus mainly on the former theory, and we give a brief explanation of the latter. 3 We assume two pure alternatives y and y; called the upper and lower benchmarks; these together with the probability grids k form the benchmark scale B k (y; y) in layer k: In Step B, pure alternatives are measured by this scale. Preferences are constructed in shallow to deeper layers, where preferences are incomplete in the beginning, except for benchmark lotteries as 1 This sounds similar to constructive decision theory in Shafer [27], [28] and in Blume et al. [4]. These authors study Savage s [26] subjective utility/probability theory so as to introduce certain constructive features for decision making. Our theory is constructive more explicitly with the introduction of probability grids and a cognitive bound. The chief di erence is that we formulate how a decision maker nds/forms his own preferences, while they add new constructs like goals or frames that shape the choices of the decision maker. 2 Our concept of probability grids may be interpreted as imprecise probabilities/similarity (cf. Augustin et al. [2], Rubinstein [25]). Imprecision/similarity is de ned as an attribute of a probability/a set of probabilities, allowing all real number probabilities. In our approach, however, probability grids in k are exact; the restriction of probabilities to k expresses imprecision in cognitive acts taken by the decision maker. 3 Our theory is dual to that in terms of certainty equivalent of a lottery (cf. Kontek-Lewandowski [20] and its references). In our method, the set of benchmark lotteries forms a base scale, while the set of monetary amounts is the base scale in the latter (see Section 4.2 in [20]). 2

4 : upper benchmark : lower benchmark Benchmark lotteries Figure 1: Step B with the benchmark scale measurement units, and in deeper layers, more precise preferences may be found. In Fig.1, the benchmark scale for layer k is depicted as the right broken line with dots; x is measured exactly by the scale; y need a more precise scale within. However, z is not done within. Two di erent roles of probability grids appear in Step E for evaluation of a lottery: (i) probability grids used for measurement of a pure alternative in Step B; (ii) probability coe cients to pure alternatives. By these, relevant cognitive depths of lotteries become more complex especially with a nite cognitive bound; this leads to incomparabilities in preferences and some violation of the EU hypothesis. This is central in our development and is closely related to the issue of bounded rationality. Let us illustrate (i) and (ii) via an example. Consider one example with the upper and lower benchmarks y; y; and the third pure alternative y with strict preferences y y y. In Step B, the decision maker looks for a probability so that y is indi erent to a lottery [y; ; y] = y (1 )y with probability for y and 1 for y; this indi erence is denoted by y [y; ; y]: (1) Suppose that this is uniquely determined as = y = : Here, exact measurement of y is successful in layer 2; where Step B is enough here. We have the other source of cognitive depths. Consider lottery d = 25 y; which includes 2 the third pure alternative y. The independence condition of the classical EU theory dictates that because of (1), [y; 83 ; y] is substituted for y in d; and d is reduced to: 2 2 y 75 d = 25 2 y 75 2 y 25 2 [y; 83 2 ; y] 75 2 y = y y: (2) Thus, y is evaluated as being indi erent to [y; 83 ; y] in Step B, but y also has a probability 2 coe cient 25 in d, which is taken into account in Step E. These steps leads to probability 2075 ; which is much more precise than either of and 25 : 2 2 As indicated in (i) and (ii), lottery d = 25 y 75 y has two types of cognitive depths; one 2 2 is simply a probability coe cient 25 and the other is 2 y = 83 from (1). Although d itself is 2 expressed as a lottery of depth 2; the total depths including these two types is 4; which is beyond the cognitive bound = 2: One point is that the resulting probability may be very precise with a relatively small cognitive bound, and the other is that this is intimately related to the EU 3

5 hypothesis. When is small, the EU hypothesis does not typically hold, while it would hold more as is getting larger. The preference formation by Steps B and E is formulated as a form of mathematical induction; Step B is the inductive base and Step E is the inductive step. Step B is spread out to layers of various depths, i.e., the induction base is spread too. These steps are described in Table 1.1: the relation D k for layer k of row B expresses preferences measured in Steps B. In layer k; % k is derived from D k and % k 1 ; the former is a part of the inductive base and the latter is the inductive step. This is a weak form of independence condition : Table 1.1 Layers 0 1 ::: k 1 k ::: B: base relations D 0 D 1 ::: D k 1 D k ::: D # # # # # E : constructed relations % 0! % 1! :::! % k 1! % k! :::! % We also provide another approach in terms of a 2-dimensional vector-valued utility functions h k i k<+1 = h[ k ; k ]i k<+1 and hu k i k<+1 = h[u k ; u k ]i k<+1 with Fishburn s [7] interval order I. In each of Steps B and E, this approach is entirely equivalent to the preference approach, depicted in Table 1.2. This may be interpreted as what von Neumann- Morgenstern [31], p.29 indicated. 4 The approaches in terms of preferences and utilities enable us to view Steps B and E in di erent ways as well as serve di erent analytic tools for studies of incomparabilities/comparabilities involved: Table 1.2 Preference theory Utility theory Step B (B0 to B3) () (Sec.3) Step B (b0 to b3) Extension (Sec.4) Extension (Sec.5) Step E (E0 to E3) () (Sec.5) Step E (e0 to e3) Our theory enjoys a weak form of the expected utility hypothesis. This will be discussed in Section 6. In the case of = 1; restricting our attention to the set of measurable pure alternatives, in Section 7, we show that our theory exhibits a form of the classical EU theory. We provide a further extension of % 1 to have the full form of classical EU theory; this extension involves some unavoidable non-constructive step, which may be interpreted as the criticism of super rationality by Simon [30]. We apply our theory to the Allais paradox, speci cally, to an experimental result from Kahneman-Tversky [14]. We show that the paradoxical results remains when the cognitive bound 3. However, when = 2; the resultant preference relation % is compatible with their experimental result, where incomparabilities play crucial roles in explaining them. A remark is on the relationship between k and exhibiting a layer and a cognitive bound. The former is a variable in our theory and the latter is a parameter of the theory. We talk about the sequences h% k i k<+1 and hu k i k<+1 describing the process of preference formation 4 Aumann [3] and Fishburn [8] considered one-way representation theorem (i.e., the only-if of (3)), dropping completeness. See Fishburn [9] for further studies. Dubra, et al. [6] developed a representation theorem in terms of utility comparisons based on all possible expected utility functions for the relation without completeness. In this literature, incomparabilities are given in the preference relation. In contrast, in our approach, incomparabilities are changing with a cognitive bound and may disappear when there are no cognitive bounds. 4

6 layer to layer up to : Nevertheless, the nal target preferences and utilities are % and u : This remark leads to the view that our theory is a generalization of the classical EU theory, which is discussed in Section 7.1. The paper is organized as follows: Section 2 explains the concept of probability grids and other basic concepts. Section 3 formulates Step B in terms preferences and utilities, and states their equivalence. Section 4 discusses Step E in terms of preferences and Section 5 does it in terms of utilities. Section 6 discusses the measurable/non-measurable lotteries, and shows that the expected utility hypothesis holds for the measurable lotteries. Section 7 discusses the connection from our theory to the classical EU theory. In Section 8, we exemplify our theory with an experimental result in Kahneman-Tversky [14]. Section 9 concludes this paper with comments on further possible studies. Proofs of all the results in each section are given in a separate subsection; only proof of Lemma 2.1 is given in Section. 2 Preliminaries Our theory is about preference formation in the context of EU theory. The classical EU theory is the reference point, but our theory deviates from it in various manners. To have clear relations between the classical EU theory and our development, we rst mention the classical theory (cf. Herstein-Milnor [12], Fishburn []), and then, we start our development. In Section 2.2, we give various basic concepts for our theory and one basic lemma. In Section 2.3, we give de nitions of preferences, indi erences, incomparabilities, and their counterparts in terms of vector-valued utility functions. 2.1 Classical EU theory Let X be a given set of pure alternatives with cardinality jxj 2. A lottery f is a function over X taking real values in [0; 1] with P x2s f(x) = 1 for some nite subset S of X: This subset S is called a support of f: We de ne L [0;1] (X) = ff : f : X! [0; 1] is a lotteryg: The set L [0;1] (X) is uncountable: We de ne compound lotteries: for any f; g 2 L [0;1] (X) and 2 [0; 1]; f (1 )g is a lottery in L [0;1] (X) de ned by (f (1 )g)(x) = f(x) + (1 )g(x) for all x 2 X. Let % E be a binary relation over L [0;1] (X); and we assume NM0 to NM2 on % E : This system is one among various equivalent systems. Axiom NM0 (Complete preordering): % E is a complete and transitive relation on L [0;1] (X): Axiom NM1 (Intermediate value): For any f; g; h 2 L [0;1] (X); if f % E g % E h; then f (1 )h E g for some 2 [0; 1]: Axiom NM2 (Independence): For any f; g; h 2 L [0;1] (X) and 2 (0; 1]; ID1: f E g implies f (1 )h E g (1 )h; ID2: f E g implies f (1 )h E g (1 )h; where the indi erence part and strict preference part of % E are denoted by E and E ; that is, f E g means f % E g & g % E f; and f E g does f % E g & not (g % E f): The following two are the key theorems in the classical EU theory. For a fruitful development of our theory, we should be conscious of how they remain in our theory. 5

7 Theorem 2.1 (Classical EU theorem). A preference relation % E satis es Axioms NM0 to NM2 if and only if there is a function u : X! R so that for any f; g 2 L [0;1] (X); where the expected utility functional E f (u) is de ned as: f % E g if and only if E f (u) E f (u); (3) E f (u) = P x2s f(x)u(x) for each f 2 L [0;1](X) with its support S: (4) Theorem 2.2 (Uniqueness up to A ne transformations). Suppose that % E satis es Axioms NM0 to NM2. If two functions u; v : X! R satisfy (3), then there are two real numbers > 0 and such that u(x) = v(x) + for all x 2 X: In these theorems, preference relation % E is given with Axioms NM0 to NM2. The theory is silent about how a decision maker nds/forms his preferences. As mentioned in Section 1, we consider this question from simple cases to more complex cases, while distinguishing between Steps B and E. In the above axiomatization, these are mixed in NM1 and NM2. We will make a clear-cut distinction between Steps B and E. In these steps, we avoid the existence of a complete preference relation dictated by Axiom NM0. Another salient restriction in our theory is on the available probabilities and is formulated by the concept of probability grids. This allows us to think about his preferences from simpler lotteries to complex ones step by step. The step-by-step consideration collapses in Axioms NM2 and NM3. The system (L [0;1] ; % E ) with Axioms NM0 to NM2 itself is not in the cetral part of our theory, but our theory is closely related to the EU hypothesis that preferences are represented by the expected utility functional E f (u). We will discuss the EU hypothesis time to time, and touch the system (L [0;1] ; % E ) only in Section Probability grids, lotteries, and decompositions Let ` be an integer with ` 2: This ` is the base for describing probability grids; we take ` = in the examples in the paper. The set of probability grids k is de ned as k = f `k : = 0; 1; :::; `kg for any nite k 0: (5) Here, 1 = f ` : = 0; :::; `g is the base set of probability grids for measurement, whereas 0 = f0; 1g is needed for technical completeness. Each k is a nite set, and 1 := [ t<1 t is countably in nite. We use the standard arithmetic rules over 1 ; sum and multiplication are needed We allow reduction by eliminating common factors; for example, is the same as 2 2 : Hence, k k+1 for k = 0; 1; ::: The parameter k is the precision of probabilities that the decision maker uses. We de ne the depth of each 2 1 by: () = k i 2 k k 1 : For example, ( 25 ) = 2 but ( 20 ) = ( ) = 1: The concept of a layer of probability grids up to a given depth k is well de ned. The decision maker thinks about his preferences along probability grids from a shallow layer to a deeper one. We use the standard equality = and strict inequality > over k : Then, trichotomy holds: for any ; 0 2 k ; either > 0 ; = 0 ; or < 0 : (6) 5 See Mendelson [22] for related basic mathematics. 6

8 Each element in k is obtained by taking the weighted sums of elements in k 1 with the equal weights: k = f `P 1 ` t : 1 ; :::; ` 2 k 1 g for any k (1 k < 1): (7) t=1 This is basic for the connection between layer k 1 to the next. A proof of (7) is not given here, but an extension will be given in Lemma 2.1 with a proof given in the Appendix. The union 1 = [ k<1 k is a proper subset of [0; 1] \ Q; where Q is the set of rational numbers. For example, when ` = ; 1 has no recurring decimals, but they are rationals: We also note that 1 depends upon the base `; for example, 1 with ` = 3 has 1 3 ; but 1 with ` = has no element corresponding to 1 3 : For any k < 1; we de ne L k (X) by L k (X) = ff : f is a function from X to k with P x2x f(x) = 1g: (8) We identify each pure alternative x with the lottery having x as its support; so X is regarded as a subset of L k (X): Speci cally, L 0 (X) = X: Since k is a nite set, every f 2 L k (X) has a nite support. Since k k+1 ; it holds that L k (X) L k+1 (X): We denote L 1 (X) = [ k<1 L k (X): As long as X is nite, L k (X) is also a nite set, but L 1 (X) is a countable set and is dense in L [0;1] (X). We de ne the depth of a lottery f in L 1 (X) by (f) = k i f 2 L k (X) L k 1 (X): We use the same symbol for the depth of a lottery and the depth of a probability. It holds that (f) = k if and only if max x2x (f(x)) = k: This is relevant in Section 6. Lottery d = 25 y 75 y 2 2 is in L 2 (X) L 1 (X) and its depth (d) = 2; but since d 0 = 20 y 80 y = y 8 y 2 L 1(X); we have (d 0 ) = 1: The decision maker thinks about preferences from shallow layers to deeper ones. This stops at a cognitive bound ; which is a natural number or in nity 1. If = k < 1; he eventually reaches the set of lotteries L (X) = L k (X); and if = 1; he has no cognitive limit; we de ne L (X) = L 1 (X) = [ k<1 L k (X): We formulate a connection from L k 1 (X) to L k (X). We say that b f = (f 1 ; :::; f`) in L k 1 (X)` = L k 1 (X) L k 1 (X) is a decomposition of f 2 L k (X) i for all x 2 X; f(x) = P` t=1 1` f t(x) and (f t (x)) (f(x)) for all t `: (9) We denote this by P` t=1 1` f t; and letting be = ( 1` ; :::; 1` ); it is written as be f: b We can regard be f b as a compound lottery connecting L k 1 (X) to L k (X) by reducing be f b to f in (9). Our theory allows only this form of a compound lotteries and reduction with the depth constraint. The next lemma states that L k (X) is generated from L k 1 (X) by taking all compound lotteries of this kind. It facilitates our induction method described in Table 1.1 reducing an assertion in layer k to layer k 1: A proof of Lemma 2.1 is given in Section : 6 Lemma 2.1 (Decomposition of lotteries). Let 1 k < 1: Then, L k (X) = ff 2 L k (X) : f has a decomposition b fg: () 6 When ` > 2; binary decompositions are not enough for Lemma 2.1, For example, consider lottery f = 3 y 3 y 4 y. This is not expressed by a binary combination of elements in L0(X) = X with weights in 1: 7

9 Furthermore, for any f 2 L k (X) with (f) > 0; there is a decomposition of b f of f so that (f t (x)) < (f(x)) for any x 2 X with f(x) > 0: (11) The right-hand side of () is the set of composed lotteries from L k 1 (X) with the equal weights: The inclusion states that the composed lotteries from L k 1 (X) belong to L k (X): The converse inclusion is essential and means that each lottery in L k (X) is decomposed to an equally weighted sum of some (f 1 ; :::; f`) in L k 1 (X)` with the depth constraint in (9). In the trivial case that f = x 2 L 0 (X) is decomposed to f b = (x; :::; x). This will be used in Proposition 4.1.(2). The latter asserts the choice of a strictly shallower decomposition for f with (f) > 0. One remark is that when f is a benchmark lottery in B k (y; y); for its decomposition b f = (f 1 ; :::; f`); each f t is a benchmark lottery in B k 1 (y; y): This fact will be used without referring. For the set of lotteries over subset X 0 of X; i.e., L k (X 0 ); we introduce the following convention. We de ne L k (X 0 ) = ff 2 L k (X) : f(x) > 0 implies x 2 X 0 g: Hence, L k (X 0 ) is a subset of L k (X): Lemma 2.1 hods for L k (X 0 ) and L k 1 (X 0 ): The lottery d = [y; 25 2 ; y] has three types of decompositions: d = t y + 5 2t [y; 5 5+t ; y] + y for t = 0; 1; 2: (12) Here, a decomposition f b = (f 1 ; :::; f ) is given as f 1 = ::: = f t = y; f t+1 = ::: = f 5 t = [y; 5 ; y] and f 5 t+1 = ::: = f = y: We use this short-hand expressions rather than a full speci cation of f b = (f 1 ; :::; f ): We should be careful about this multiplicity. The reason for explicit considerations of layers for L k (X) and also preference relation % k is to avoid collapse from a layer to a shallower one. Without them, we may have a di culty in identifying the sources for preferences. For example, the weighted sum 5 [ 25 y 75 y] [ 75 y 2 25 y] is reduced to 5 2 y 5 5 y; preferences about y 5 y may possibly come from layer 2 or from layer 0: To prohibit such collapse, we take explicitly depths of layers into account in (9). 2.3 Incomplete preference relations and vector-valued utility functions We consider two methods to represent the decision maker s desires: a preference relation and a utility function. We starts with incomplete preferences and, correspondingly, representing utility functions become vector-valued with the interval order. These are rst departures from the classical EU theory. Let % be a preference relation over a given set, say A: For f; g 2 A; the expression f % g means that f is strictly preferred to g or is indi erent to g: We de ne the strict (preference) relation ; indi erence relation ; and incomparability relation 1 by f g if and only if f % g and not g % f; (13) f g if and only if f % g and g % f; f 1 g if and only if neither f % g nor g % f: All the axioms are given on the relations %; ; ; and the relation 1 is de ned as the residual part of %. Although and 1 are sometimes regarded as closely related (cf. Shafer [27], p.469), they are well separated in Theorem 6.2 in our theory. 8

10 In the classical theory in Section 2.1, the preference relation % E is assumed to be complete. Since, however, we consider a formation of preferences, our theory should avoid this completeness assumption. Nevertheless, it appears as a result when a domain of lotteries is restricted. Another method of measurement of desires is by a vector-valued function u with the interval order introduced by Fishburn [7]. Let u(f) = [u(f); u(f)] be a 2-dimensional vector-valued function from its domain A to the set Q 2 = Q Q with u(f) u(f) for each f 2 A: The components u(f) and u(f) are interpreted as the least upper and greatest lower bounds of possible utilities from f: We say that u(f) is e ectively single-valued i u(f) = u(f); in this case, we write u(f) = u(f) = u(f); dropping the upper and lower bars. We use the interval order I over the values of u; for f; g 2 A; u(f) I u(g) if and only if u(f) u(g): (14) That is, f and g are ordered if and only if the greatest lower bound u(f) from f is larger than or equal to the least upper bound u(g) from g: This I allows incomparabilities, for example, if u(f) = [ 9 ; 7 83 ] and u(g) = [ ; 83 ]; then f and g are incomparable by 2 2 I : The relation I is transitive, but (u(f) I u(g) & u(g) I u(f)) is equivalent to u(f) = u(f) = u(g) = u(g); i.e., this is the case only when the values u(f) and u(g) are e ectively single-valued and identical. 3 Measurement Step We formulate Step B of measurement of pure alternatives up to cognitive bound : This has two sides: in terms of preference relations hd k i k<+1 and in terms of vector-valued utility h k i k<+1. We show the representation theorem on hd k i k<+1 by h k i k<+1 ; and the uniqueness theorem on h k i k<+1 up to positive linear transformations. Finally, we mention that these are well interpreted in terms of Simon s [29] satis cing/aspiration argument. 3.1 Base preference streams The set of pure alternatives X is assumed to contain two distinguished elements y and y; which we call the upper and lower benchmarks. Let k < 1: We call an f 2 L k (X) a benchmark lottery of depth (at most) k i f(y) = and f(y) = 1 for some 2 k ; which we denote by [y; ; y]: The benchmark scale of depth k is the set B k (y; y) = f[y; ; y] : 2 k g: In particular, B 0 (y; y) = fy; yg: The dots in Fig.1 express the benchmark lotteries. We de ne B 1 (y; y) = [ k<1 B k (y; y): The depth of a benchmark lottery [y; ; y] is determined to be the depth of ; i.e., ([y; ; y]) = (): We denote a cognitive bound by ; which is a natural number or = 1: We use k as a variable expressing a natural number of a layer within the theory, but as a constant parameter of it. Stipulating = 1; k < + 1 expresses the two statements k if < 1 and k < if = 1. This constant plays an active role as a small constraint such as = 2 or 3 in Example 5.1 and Section 8, and as = 1 in Section 7 for consideration of the expected utility hypothesis. Let D k be a subset of D k = B k (y; y) 2 [ f(x; g); (g; x) : x 2 X and g 2 B k (y; y)g: (15) 9

11 Thus, D k consists of the scale part of the benchmarks and the measurement part of pure alternatives. The scale part allows the decision maker to make comparisons between any grids of depth k. For a pure alternative x 2 X; he thinks about where x is located in the benchmark scale B k (y; y); it may or may not correspond to a grid, which is seen in Fig.1. For example, if (x; g) 2 D k but (g; x) =2 D k ; then x is strictly better than the grid g; and if (x; g) =2 D k and (g; x) =2 D k ; then x and g are incomparable for him. We make four axioms on hd k i k<+1. Axiom B0 requires pure alternatives be between the upper and lower benchmarks y; y: Axiom B0 (Benchmarks): y D 0 x and x D 0 y for all x 2 X: The next states that preferences over B k (y; y) are the same as the natural order on k : Axiom B1 (Benchmark scale): For ; 0 2 k ; [y; ; y] D k [y; 0 ; y] if and only if 0 It follows from Axiom B1 that for ; 0 2 k ; [y; ; y] B k [y; 0 ; y] if and only if > 0 : (16) Also, = 0 if and only if [y; ; y] and [y; 0 ; y] are indi erent: Thus, D k is a complete relation over B k (y; y) by (6). This is the scale part of D k ; and is precise up to k : Since y = [y; 1; y] and y = [y; 0; y]; it follows from (16) that y B;0 y: Measurement is required to be coherent with the scale part given by Axiom B1. Axiom B2 (Monotonicity): For all x 2 X and ; 0 2 k ; if [y; ; y] D k x and 0 > ; then [y; 0 ; y] B k x; and if x D k [y; ; y] and > 0 ; then x B k [y; 0 ; y]: This implies no reversals with Axiom B1; if [y; ; y] D k x and x D k [y; 0 ; y]; then 0 : Indeed, if < 0 ; then [y; 0 ; y] B k x by B2, which implies not x D k [y; 0 ; y]: If we assume transitivity for D k over D k ; B2 could be derived from B1, but we adopt B2 instead of transitivity, since B2 gives a more speci c property to the measurement step. The last requires the preferences in layer k be preserved in the next layer k + 1: This is expressed by the set-theoretical inclusion in Table 1.1. Axiom B3 (Preservation): For all f; g 2 D k ; f D k g implies f D k+1 g: The above axioms still allow great freedom for base preference relations hd k i k<+1 : To see this fact as well as how the measurement step B of utilities from pure alternatives goes on, we consider vector-valued utility functions with the interval order I in Section Base utility streams We consider another way of Step B in terms of vector-valued utility functions with the interval order I. Let h k i k<+1 = h[ k ; k ]i k<+1 be a sequence of vector-valued functions so that for each k < + 1; k is a function from B k (y; y) [ X to Q 2 such that k (f) k (f) for all f 2 B k (y; y) [ X: Recall that when k (f) is e ectively single-valued, we write k (f) = k (f) = k (f). The following conditions on h k i k<+1 are not exactly parallel to the axiomatic system B0 to B3, but these two systems are equivalent, which is stated in Theorem 3.1. We de ne a base (upper-lower) utility stream h k i k<+1 = h[ k ; k ]i k<+1 by b0 to b3 : b0: 0 (y) > 0 (y);

12 and for k < + 1; b1: k ([y; ; y]) = k (y) + (1 ) k (y) for all [y; ; y] 2 B k (y; y); b2: for each x 2 X; k (x) = k ([y; x ; y]) and k (x) = k ([y; x ; y]) for some x and x in k ; b3: for each x 2 X; k (x) k+1 (x) k+1 (x) k (x): Condition b0 xes the utility values from the upper and lower benchmarks y and y; which corresponds to the implication of B0. Then, b1 means that for benchmark lotteries [y; ; y] 2 B k (y; y); k ([y; ; y]) is e ectively single-valued and takes the expected utility value of y and y; which corresponds to B1. Here, the EU hypothesis is included. b2 states that the least upper and greatest lower utilities of x 2 X are measured by the benchmark scale B k (y; y); this does not exactly correspond to B2, but it does an implication of B2 with the help of transitivity for k implied by the interval order I. Corresponding to B3, b3 states that k (x) and k (x) are getting more accurate as k increases. These imply k (y) = 0 (y) and k (y) = 0 (y) for any k < + 1: (17) Now, we have Theorem 3.1. As stated, all proofs are given in separate subsections. Theorem 3.1 (Representation for Step B). A base preference stream hd k i k<+1 satis es Axioms B0 to B3 if and only if there is a base utility stream h k i k<+1 satisfying b0 to b3 such that for any k < + 1 and (f; g) 2 D k ; f D k g if and only if k (f) I k (g): (18) Although k is vector-valued, it satis es the EU hypothesis, since by b1; k ([y; ; y]) = k (y) + (1 ) k (y) for [y; ; y] 2 B k (y; y); and by b2; k (x) = x k (y) + (1 x ) k (y) and k (x) = x k (y) + (1 x ) k (y) for x 2 X: The EU hypothesis does not hold for the representation theorem for Step E (Theorem 5.1). We have the uniqueness theorem. Theorem 3.2 (Uniqueness). Let hd k i k<+1 satisfy Axioms B0 to B3. If h k i k<+1 and h 0 k i k<+1 satisfying b0 to b3 represent hd k i k<+1 in the sense of (18), there are rational numbers > 0 and such that 0 k (x) = k(x)+ = [ k (x)+; k (x)+] for all x 2 X and k < +1: Conditions b0 to b3 require k (x) = ( k (x); k (x)) be represented essentially by two values and 0 in k with 0 (y) and 0 (y): However, 0 (y) and 0 (y) for each h k i k<+1 are allowed to take any two rational numbers in Q only with 0 (y) > 0 (y): This is the reason for the above uniqueness result. The uniqueness up to a positive linear transformation plays a crucial role in the literature of bargaining theories such as Nash [23] and the Nash welfare function theory (Kaneko- Nakamura [18]). The rational number scalars are enough for the 2-person case and the real-algebraic numbers are enough for the general n-person case (cf. Kaneko [15] for the 2-person case). It is easy to generalize Theorem 3.2 for the real numbers scalars, but the problem is how much we restrict the scalors. Theorem 3.2 is suggestive of how bounded rationality is incorporated to these theories. The processes described in terms of hd k i k<+1 and/or h k i k<+1 are thought experiments by the decision maker to search preferences/utilities in his mind. From the viewpoint of bounded 11

13 Figure 2: upper and lower utility functions rationality, he may stop his search when he is satis ed and/or is already tired. This is the same as Simon s [29] argument of satis cing/aspiration. First, we consider Example 3.1, and then we exemplify the satis cing/aspiration argument. Example 3.1. Let X = fy; y; yg; 0 (y) = [1; 1]; 0 (y) = [0; 0]; and 0 (y) = [1; 0]: Also, let 1 (y) = [ 9 ; 7 ]: Then, 1(f) = [ 8 ; 8 8 ] for f = [y; ; y] by b1: Then 1(y) I 1 (f) and 1 (f) I 1 (y); so y and f are incomparable with respect to D 1 by (18). In Fig.2, h k (y)i k<+1 = h[ k (y); k (y)]i k<+1 is described as solid lines in cases A, B, and C. Since 0 (y) = [1; 0]; we have y B 0 y B 0 y by (18). For k = 2; in A, 2 (y) = [ 77 ; 77 ] and the 2 2 decision maker prefers f = [y; 8 ; y] to y; and in B, 2(y) = [ 83 2 ; 83 k (y) = [ 9 ; 7 ]; he prefers y to f: In C, 2 ] is constant for k 2; he gives up comparisons between y and f after k = 1: Introspection Process of Simon s satis cing/aspiration: The decision maker starts evaluating of a pure alternative y 2 X with the benchmark scale B 0 (y; y): Suppose that he nds 0 (y) = [1; 0]; i.e., he attaches the upper value 1 and lower value 0 to y: If = 0; his introspection is over. Let 1: Then, he goes to layer 1 and uses the more precise scale B 1 (y; y) to measure y: In Example 3.1, y is better than [y; 7 9 ; y] but worse than [y; ; y]: Still, he has not reached a very precise measurement. If = 1; he stops introspection. Let 2: Then, he goes to layer k = 2; in A of Fig.2, he reaches precise utility values 2 (y) = [ 77 ; 77 ]; but in C, he has 2 2 still imprecise values 2 (y) = [ 9 ; 7 ] and does not improve them any more even for k > 2. In case A, there are several possible interpretations: one is that 2 (y) = [ 77 ; 77 ] expresses 2 2 his preferences precisely, and the other is that he is still unsure about the value of y; for example, the lower and upper values may be 77 and 78 ; but according his aspiration level, the di erence = 1 is tiny and he does not care about the choice between 77 and 78 : By chance, he chooses 2 (y) = [ 77 ; 77 ]: In case C, (y) = [ 9 ; 7 ] is good enough for him, and he forgets further updating. Thus, exact values may include some imprecision induced by his aspiration level. This is an attribute of cognitive acts by the decision maker, instead of an attribute of probabilities. 12

14 3.3 Proofs Proof of Theorem 3.1. (If): Suppose that h k i k<+1 satis es b0 to b3 and that (18) holds for hd k i k<+1 and h k i k<+1 B0: We have, by b0; b1 and (17), 0 (y) 0 (x) and 0 (x) 0 (y); i.e., y D 0 x D 0 y: Thus, B0. B1: By (18), b1, and (17), we have [y; ; y] D k [y; 0 ; y] if and only if k ([y; ; y]) I k ([y; 0 ; y]) if and only if 0 (y) + (1 ) 0 (y) 0 0 (y) + (1 0 ) 0 (y) if and only if 0 : That is, B1. B2: Let [y; ; y] D k x and 0 > : By b2 and (18), we have 0 k (y) + (1 0 ) k (y) > k (y)+ (1 ) k (y) k (x): Thus, by b1; k ([y; 0 ; y]) = 0 k (y) + (1 0 ) k (y) > k (x): By (18), we have [y; 0 ; y] B k x: The other case is symmetric. B3: Let f D k g: By (18), we have k (f) k (g): Let f = x 2 X and g = [y; ; y] 2 B k (y; y): Then, k (g) = k+1 (g) by b1: Then, by b3, we have k+1 (f) k (f) k (g) = k+1 (g): By (18), f D k+1 g: The case f 2 B k (y; y); g = x 2 X is parallel. The case f = [y; ; y], g = [y; 0 ; y] 2 B k (y; y) is similar. (Only-if): Suppose that hd k i k<+1 satisfying Axioms B0 to B3 is given. We construct a base utility stream h k i k<+1 satisfying (18). We de ne [ k ; k ]; k < + 1; as follows: for any f 2 B k (y; y) [ X; k (f) = minf 2 k : [y; ; y] D k fg; (19) k (f) = maxf 2 k : f D k [y; ; y]g: It holds that k (f) = k (f) for f 2 B k (y; y): Consider f = [y; f ; y]; g = [y; g ; y] 2 B k (y; y): Then, f D k g if and only if [y; f ; y] D k [y; g ; y] if and only if f g ; i.e., k (f) I k (g) by B1. Let f 2 B k (y; y) and g = x 2 X: Denote k (f) = f and k (x) = x : Suppose f D k x: By (19), [y; f ; y] = f D k [y; x ; y]: By B1, f x ; i.e., k (f) I k (x): The converse is obtained by tracing this back. Thus, f D k x if and only if k (f) I k (x): The case f = x 2 X; g 2 B k (y; y) is parallel. By (16) and B0, we have b0: By (19), we have b2 and b3: Consider b1: Since k (y) = 1 and k (y) = 0; the set f k (f) : f 2 B k (y; y)g is the same as k : For any f = [y; ; y]; g = [y; 0 ; y] 2 B k (y; y); we have, by (17), k (f) > k (g) if and only if f B k g if and only if > 0 : Hence, k (f) = = k (y) + (1 ) k (y), which is b1: Proof of Theorem 3.2. Let = ( 0 0 (y) 0 0 (y))=( 0(y) 0 (y)) and = ( 0 (y) 0 0 (y) 0 0 (y) 0(y))=( 0 (y) 0 (y)): Noting (17), we have 0 k (y) = k(y) + and 0 k (y) = k(y) +. For any [y; ; y] 2 B k (y; y); we have 0 k ([y; ; y]) = k(y)+ (1 ) k (y) = k ([y; ; y]) + by b1. For any x 2 X; we have x and x in k by b2 for k such that k (x) = [ k ([y; x ; y]); k ([y; x ; y])]: Let 0 x and 0 x be given by b2 for 0 k : Suppose 0 x 6= x ; say, x > 0 x: Then, k ([y; x ; y]) I k (x); but k (x) = k ([y; 0 x; y]) > k ([y; 0 x; y]): Hence, k ([y; 0 x; y]) I k (x): However, by de nition of 0 x; we have 0 k ([y; 0 x; y]) I 0 k (x): This is impossible since k and 0 k represent the same D k : The case x < 0 x is parallel. Thus, 0 x = x ; and similarly, 0 x = x ; which imply 0 k (x) = [0 k ([y; x; y]); 0 k ([y; x; y])] = [ k ([y; x ; y]); k ([y; x ; y])] + = k (x) + : 13

15 4 Extension Step: Extended Preference Streams Step B is an introspective process to nd preferences hidden in the mind of the decision maker. On the other hand, Step E is a logical process to extend base preferences found in Step B. This di erence may create some logical di culty in adopting the standard method of representing preferences in terms of a binary relation. We actually show that we can avoid this di culty. Keeping this remark in mind, we present our axiomatic system for Step E. Throughout this section, let hd k i k<+1 be a given base preference stream satisfying Axioms B0 to B Extended preference stream Here, we consider how D k is extended to L k (X) for k < + 1: Axiom E0 is to convert base preferences D k to % k for each k < + 1; depicted as the vertical arrows in Table 1.1. Axiom E0 (Extension)(i): For any (f; g) 2 D 0 ; f D 0 g if and only if f % 0 g: (ii): For any k (1 k < + 1) and (f; g) 2 D k ; if f D k g; then f % k g: This is the ultimate source for preferences for Step E. For k = 0; the base preferences are only the direct source for % 0 : For k 1; in addition to the base preferences, there is another source from the previous % k 1 ; thus, (ii) has only one direction. However, we will show that as long as the domain D k is concerned, the converse of (ii) holds for our intended preference stream h% k i k<+1. Consider the connection between layers k 1 and k: For f b = (f 1 ; :::; f`) and bg = (g 1 ; :::; g`); we write f b % k bg i f t % k g t for all t = 1; :::; `: Recall that a decomposition of f 2 L k (X) is de ned by (9): We formulate a derivation of % k from % k 1 as follows: let 1 k < + 1: Axiom E1 (Derivation from the previous layer): Let f 2 L k (X); g 2 B k (y; y); and b f; bg their decompositions: If b f % k 1 bg or bg % k 1 b f; then f %k g or g % k f; respectively. In layer k 1; each f t of f b = (f 1 ; :::; f`) is compared with the corresponding benchmark lottery g t. These preferences are extended to layer k. In Table 1.1, the horizontal arrows indicate this derivation. When = 2; the lottery d = 25 y 75 y in the example of (2) should be evaluated in 2 2 terms of lotteries in B 2 (y; y): Axiom NM2 (Independence) is much stronger in that comparisons jump from one layer to a layer of any depth. In E2, a connection from one layer to the next with equal weights describes the step-by-step extension of preferences by the decision maker. The preferences derived by the above axioms are extended by transitivity: let 0 k < + 1: Axiom E2 (Transitivity): For any f; g; h 2 L k (X); if f % k g and g % k h; then f % k h: Here, we regard Axioms E0 to E2 as inference rules, rather than properties to be satis ed by % k. This means that the decision maker constructs % 0 ; % 1 ; :::; step by step, using these axioms. As mentioned above, this view may involve some di culty; it is logically possible that Axioms E0 to E2 may lead to new unintended preferences. Theorem 4.1 states that this is not the case for the constructed preferences. These also have the following additional conditions: E0 : for all k < + 1 and (f; g) 2 D k ; f D k g if and only if f % k g; E1 : E1 holds and if the premise of E1 includes strict preferences, so does the conclusion. Condition E0 states that h% k i k<+1 is a faithful extension of hd k i k<+1 in that as long as a 14

16 pair of lotteries in D k is concerned; the extended relation % k has no super uous preferences. E1 is a strengthening of E1, too. Without these, some preferences would be added in the derivation process of % 0 ; % 1 ; :::; and also we would have inconveniences in applications. Note that E2 (transitivity) preserves strict preferences in the same way as E1 : To prove that our constructive extended stream h% k i k<+1 enjoys E0, E1 ; and E2, we rst show the following lemma using the EU hypothesis. However, this di ers from the intended stream, which enjoys the EU hypothesis only partially. For this lemma, a base utility stream h k i k<+1 satisfying (18) given in Theorem 3.1 is used. Lemma 4.1 (Consistency of E0, E1, and E2). There is a stream of binary relations h% k i k<+1 satisfying Axioms E0 ; E1, and E2. One of such a h% k i k<+1 is given as follows: for all k < + 1; we de ne % k by f % k g if and only if E f ( k ) E g ( k ): (20) A point of this lemma is that we have no problem in regarding E0 to E2 as requirements for binary relations h% k i k<+1; and also, some satis es E0 and E2 : Now, we prepare a few concepts for the main theorem, i.e., Theorem 4.1, of this section: Let h% k i k<+1 be a stream satisfying E0 to E2. We say that h% k i k<+1 is the smallest stream i for any h% 0 k i k<+1 satisfying E0 to E2, and f; g 2 L k (X); k < + 1; f % k g implies f % 0 k g: (21) Also, the set of preferences over L k (X) derived from % k 1 by E1 is denoted by (% k 1 ) E1 ; and the set of transitive closure of F L k (X) 2 is denoted by F tr ; i.e., (f; g) 2 F tr if and only if there is a nite sequence f = h 0 ; h 1 ; :::; h m = g such that (h t ; h t+1 ) 2 F for t = 0; :::; m 1: Theorem 4.1 (Smallest extended stream). The sequence h% k i k<+1 of the sets generated by the following induction: % 0 = (D 0 ) tr ; and % k = [(% k 1 ) E1 [ (D k )] tr for each k (1 k < + 1) (22) is the smallest stream satisfying E0 to E2. Also, h% k i k<+1 satis es E0 and E1 : The construction starts with % 0 = (D 0 ) tr ; which is well de ned since D 0 is a binary relation in D 0 : Then, provided that % k 1 and D k are already given, % k is de ned to be [(% k 1 ) E1 [(D k )] tr : This is a subset of L k (X) 2 ; thus, it is a binary relation. This preference stream is unique, and is the smallest among the streams satisfying E0 to E2. Furthermore, the constructed stream satis es E0 and E1 : We extract the essential addition in (22) to Axioms E0 to E2 and formulate it as Axiom E3. It states that a preference f % k g is based on comparisons with the benchmark scale B k (y; y) with either D k or % k 1 : We note that h in E3 may be the same as f or g: Axiom E3 (i): For any f; g 2 L 0 (X); if f % 0 g; then f % 0 h % 0 g for some h 2 B 0 (y; y): (ii): For any f; g 2 L k (X) (k 1); if f % k g; then there is an h 2 B k (y; y) with f % k h % k g such that for the rst pair (f; h); f D k h holds or f; h have decompositions f; b b h with b h % k and the same holds for the second pair (h; g): The key of Axiom E3 is to include the depth constraint; comparison f % k g e ectively comes from the benchmark scale B k (y; y) of the same depth k: This constraint with a nite cognitive 15 1 bg;

17 bound makes the EU hypothesis hold partially. The preference stream given by (20) of Lemma 4.1 enjoys the EU hypothesis, but since it does not take the depths of f; g into account, Axiom E3 is violated. This violation will be seen in Example 5.1. The stream h% k i k<+1 given by (22) is characterized by adding E3 to E0 to E2. Theorem 4.2 (Uniqueness by E0 to E3). Any extended stream satisfying E0 to E3 is the same as the preference stream h% k i k<+1 given by Theorem 4.1. Throughout the following, the stream given by (22) is denoted by h% k i k<+1 : Other streams may have some additional superscripts such as 0 ; : Proposition 4.1 will be used in the subsequent analyses: (1) is the horizontal arrows in Table 1.1, and (2) that % k is bounded in L k (X) by the upper and lower benchmarks y and y: Proposition 4.1. Let h% k i k<+1 satis es E0 to E3, and 1 k < + 1: (1)(Preservation of preferences): For any f; g 2 L k 1 (X); f % k 1 g implies f % k g: (2): y % k f % k y for any f 2 L k (X): Remark 4.1 (Partial EU hypothesis in the two systems). Axiom B1 and b1 assume the EU hypothesis along the benchmark scale B k (y; y); and E2 is a very weak form of Axiom NM2 (independence). The other axioms are related to it only in that preferences are considered through comparisons with B k (y; y). As mentioned above, Axiom E3 includes the depth constraint on preference comparisons; it follows from Lemma 4.1 and Theorem 4.1 that there are possibly many preference streams satisfying E0 to E2; among which the EU representation in (20) is allowed. A departure from the EU hypothesis is caused by two types of depths included in a lottery and their interactions with a cognitive bound < 1: For example, lottery d = 25 y 75 y involves the depths of coe cient 25 and of evaluation y. This and E3 make the EU hypothesis hold only for some partial domain, which is explicitly studied in Section Proofs Proof of Lemma 4.1. We show that h% k i k<+1 given by (20) satis es E0 ; E1 ; and E2. By Theorem 3.2, we can assume that k (y) = 1 and k (y) = 0: Since E f ( k ) = if f = [y; ; y] 2 B k (y; y) and E x ( k ) = k (x) if f = x 2 X: Hence, by (18) and b2; f D k x if and only if k (x) if and only if E f ( k ) E x ( k ): The other cases are symmetric. Thus, E0 holds for any (f; g) 2 D k : It remains to show that h% k i k<+1 satis es E1 and E2. Since (20) gives the interval order over the set f[e f ( k ); E f ( k )] : f 2 L k (X)g; E2 holds. We show E1 : Let f 2 L k (X); g 2 B k [y; y] and their decompositions f b and bg with f b % k 1 bg: By (20), E f t ( k ) E gt ( k ) for all t = 1; :::; `: Then, E f ( k ) = E be f b( k ) = P` t=1 1` Ef t ( k ) P` t=1 1` Eg t ( k ) = E bebg ( k ) = E g ( k ): If strict preferences are included in the decompositions, the conclusion is strict; thus we have E1 : Proof of Theorem 4.1. This has the three assertions: (a) h% k i k<+1 is a sequence of a binary relations satisfying Axioms E0 to E2; (b) it is the smallest in the sense of (21) among streams h% 0 k i k<+1 satisfying E0 to E2; and (c) E0 ; E1 hold for h% k i k<+1 : (a): E2 follow directly from (22). Consider E0. (ii) follows from (22). We show that % 0 = (D 0 ) tr satis es that for any (f; g) 2 D 0 ; f % 0 g implies f D 0 g: Since % 0 = (D 0 ) tr ; there is a sequence f = h 0 D 0 ::: D 0 h m = g. If h t 2 X B 0 (y; y); then h t 1 2 B 0 (y; y) and h t+1 2 B 0 (y; y): By B2, t 1 t+1 ; where h t 1 = [y; t 1 ; y] and h t+1 = [y; t+1 ; y]: If h t ; h t+1 2 B 0 (y; y); then 16

18 t t+1 : Hence, we can shorten the sequence to f = h 0 D 0 h m = g: Thus, f D 0 g: Consider E1. Suppose that f 2 B k [y; y] and g 2 L k (X) have decompositions f; b bg 2 L k 1 (X) with f b % k 1 bg: By (22), we have f = e f b % k e bg = g: The symmetric case, bg % k 1 f; b is similar. (b): We prove by induction on k that h% k i k<+1 satis es (21) for any h% 0 k i k<+1 satisfying E0 to E2. When k = 0; we have % 0 = (D 0 ) tr by (22): Let f % 0 g; i.e., f (D 0 ) tr g; which implies that there is a sequence f = h 0 D 0 h 1 D 0 ::: D 0 h m = g: By E0.(i), we have f = h 0 % 0 0 h 1 % 0 0 ::: % 0 0 h m = g: By E2 for % 0 0 ; we have f %0 0 g. Now, we assume that (21) holds for k 1: Let f % k g: By (22), there is a sequence f = h 0 % k ::: % k h m = g such that each h t % k h t+1 is a consequence of E1 or h t % k h t+1 is h t D k h t+1 : In the rst case, there are decompositions b h t ; b h t+1 of h t ; h t+1 such that b h t % k 1 b ht+1 : By the induction hypothesis, we have b h t % 0 b k 1 h t+1 : Thus, h t % 0 k h t+1 by E1 for % 0 k : In the second case, h t D k h t+1 implies h t % 0 k h t+1 by E0.(ii) for % 0 k : Hence, f %0 k g by E2 for %0 k : (c): Take h% k i k<+1 given by Lemma 4.1. Since h% k i k<+1 satis es E0 to E2, it holds that for all k < + 1 and f; g 2 L k (X); f % k g implies f % k g: (23) E0 : Since E0 holds for % k by Lemma 4.1, we have: for any (f; g) 2 D k; f % k g implies f D k g; thus, f % k g implies f D k g: The converse is from (22). E1 : Let f; g 2 L k (X): Let f; b bg be decompositions of f; g so that f b % k 1 bg with strict preferences for some components. Hence, by (23), the same holds for % k 1 : Hence, by E2 for % k 1 ; we have f k g: In this case, g % k f is impossible; if it was the case, we would have, by (23), f k g; a contradiction. Hence, f k g: Proof of Theorem 4.2. Let h% k i k<+1 be any extended stream satisfying E0 to E3. We prove by induction on k < + 1 that for any f; g 2 L k (X); f % k g if and only if f % k g: (24) Since h% k i k<+1 is the smallest stream satisfying E0 to E2 by Theorem 4.1, the if part holds for any k < + 1: Consider the only-if part. Let k = 0: Let f; g 2 L 0 (X) with f % 0 g: Then, by E3.(i), we have an h 2 B 0 (y; y) with f % 0 h % 0 g: But this h is either y or y: If h = y; then h D 0 f and f D 0 h by B0: Since h D 0 g by B0, we have f (D 0 ) tr g; i.e., f % 0 g by (22). The case h = y is similar. We make the induction hypothesis that the only-if part holds for k 1: Let f % k g: Then, by E3, we have h 0 := f % k h 1 % k h 2 := g for some h 1 2 B k (y; y): If h 0 = x 2 X; then, h 0 = x has no decomposition; thus, by E3.(ii), h 0 D k h 1 ; which implies h 0 % k h 1 by E0 for % k. Let h 0 =2 X: By E3.(ii), b h 0 % b k 1 h 1 for some decompositions b h 0 ; b h 1 of h 0 ; h 1 : By the induction hypothesis, we have b h 0 % k 1 b h1 : Thus, by E1 for % k ; we have h 0 % k h 1 : By the same argument, we have h 1 % k h 2 : Thus, by (22), h 0 % k h 2 ; i.e., f % k g: Proof of Proposition 4.1. (1): Let f 2 L k 1 (X) and g 2 B k 1 (y; y): Suppose f % k 1 g: Then, f; g 2 L k 1 (X) L k (X): Let f 1 = ::: = f` = f and g 1 = ::: = g` = g: Then, f = P` t=1 1` f t and f = P` t=1 1` g t: By E1, we have f % k g: The case g % k 1 f is similar. Let f; g 2 L k 1 (X) with f % k 1 g: Then, by E3.(ii) for k; f % k 1 h % k 1 g for some h 2 B k 1 (y; y): It follows from the conclusion of the above paragraph that f % k h % k g: By E2, we have f % k g: (2): Let f 2 L 0 (X) = X: By B0 and % 0 = D 0, we have the assertion for k = 0. Suppose the 17